Data

The data you will be using comes from the Assistments online intelligent tutoring system (https://www.assistments.org/). It describes students working through online math problems. Each student has the following data associated with them:

Start by uploading the data

D1 <- read.csv("Assistments-confidence.csv")
D1$id <- NULL
#We won't need the id variable, so remove that.

Create a correlation matrix of the relationships between the variables, including correlation coefficients for each pair of variables/features.

#You can install the corrplot package to plot some pretty correlation matrices (sometimes called correlograms)

library(corrplot)
## corrplot 0.84 loaded
#Generate pairwise correlations
COR <- cor(D1)

corrplot(COR, order="AOE", method="circle", tl.pos="lt", type="upper",        
tl.col="black", tl.cex=0.6, tl.srt=45, 
        addCoef.col="black", addCoefasPercent = TRUE,
        sig.level=0.50, insig = "blank")

#Study your correlogram image and save it, you will need it later

Create a new data frame with the mean_correct variables removed

D2 <- D1[, -c(4)]

#The, scale and center your data for easier interpretation
D2 <- scale(D2, center = TRUE)

Now run the PCA on the new data frame

pca <- prcomp(D2, scale = TRUE)

Although princomp does not generate the eigenvalues directly for us, we can print a list of the standard deviation of the variance accounted for by each component.

pca$sdev
## [1] 1.2825140 1.0543565 1.0245688 0.9621486 0.8556715 0.7320146
#To convert this into variance accounted for we can square it, these numbers are proportional to the eigenvalue

pca$sdev^2
## [1] 1.6448423 1.1116675 1.0497412 0.9257299 0.7321737 0.5358454
#A summary of our pca will give us the proportion of variance accounted for by each component

summary(pca)
## Importance of components:
##                           PC1    PC2    PC3    PC4    PC5     PC6
## Standard deviation     1.2825 1.0544 1.0246 0.9621 0.8557 0.73201
## Proportion of Variance 0.2741 0.1853 0.1750 0.1543 0.1220 0.08931
## Cumulative Proportion  0.2741 0.4594 0.6344 0.7887 0.9107 1.00000
#We can look at this to get an idea of which components we should keep and which we should drop

plot(pca, type = "lines")

Decide which components you would drop and remove them from your data set.

Comments: Referring to the plot and PCA, PC3, PC4, PC5, PC6 will be dropped.

Part II

#Now, create a data frame of the transformed data from your pca.

D3 <- pca$x[, c(1,2)]
#Attach the variable "mean_correct" from your original data frame to D3.

D4 <- cbind(D3, as.data.frame(D1$mean_correct))

#Now re-run your scatterplots and correlations between the transformed data and mean_correct. If you had dropped some components would you have lost important infomation about mean_correct?

COR2 <- cor(D4)

corrplot(COR2, order="AOE", method="circle", tl.pos="lt", type="upper",        
tl.col="black", tl.cex=0.6, tl.srt=45, 
        addCoef.col="black", addCoefasPercent = TRUE,
        sig.level=0.50, insig = "blank")

## Now print out the eigenvectors (often called loadings) for the components you generated:

pca$rotation
##                               PC1         PC2         PC3        PC4
## prior_prob_count      -0.26034140  0.45818753 -0.40090679 -0.6897642
## prior_percent_correct  0.16840319  0.81617867  0.09267306  0.2640040
## problems_attempted    -0.45568733  0.31685183  0.36387724  0.3168141
## mean_hint             -0.63337594 -0.12501620 -0.08008842 -0.1122586
## mean_attempt          -0.54200011 -0.08510858 -0.04585364  0.3108682
## mean_confidence        0.03581325  0.02547483 -0.83051917  0.4948890
##                                PC5         PC6
## prior_prob_count      -0.007142834 -0.29280482
## prior_percent_correct  0.298843852  0.37134715
## problems_attempted    -0.592336569 -0.32911025
## mean_hint             -0.102302115  0.74412634
## mean_attempt           0.697232132 -0.33781385
## mean_confidence       -0.251357022 -0.01452143
#Examine the eigenvectors, notice that they are a little difficult to interpret. It is much easier to make sense of them if we make them proportional within each component

loadings <- abs(pca$rotation) #abs() will make all eigenvectors positive

sweep(loadings, 2, colSums(loadings), "/") #sweep() computes each row as a proportion of the column. (There must be a way to do this with dplyr()?)
##                              PC1        PC2        PC3        PC4
## prior_prob_count      0.12423113 0.25081186 0.22101700 0.31516257
## prior_percent_correct 0.08035956 0.44677621 0.05108998 0.12062699
## problems_attempted    0.21744737 0.17344469 0.20060288 0.14475664
## mean_hint             0.30223780 0.06843387 0.04415217 0.05129246
## mean_attempt          0.25863458 0.04658844 0.02527878 0.14203987
## mean_confidence       0.01708956 0.01394492 0.45785919 0.22612148
##                               PC5        PC6
## prior_prob_count      0.003664468 0.14011651
## prior_percent_correct 0.153315014 0.17770154
## problems_attempted    0.303884750 0.15748983
## mean_hint             0.052483764 0.35608836
## mean_attempt          0.357699023 0.16165478
## mean_confidence       0.128952980 0.00694897
#Now examine your components and try to come up with substantive descriptions of what some might represent?

### Comments: PC1 demonstrates students' current status in the session, and it is mostly influenced by problems_attempted, mean_hint and mean_attempt. PC2 demonstrates students' status in the previous sessions, which is mostly influenced by prior_prob_count and prior_percent_correct.

#You can generate a biplot to help you, though these can be a bit confusing. They plot the transformed data by the first two components. Therefore, the axes represent the direction of maximum variance. Then mapped onto this point cloud are the original directions of the variables, depicted as red arrows. It is supposed to provide a visualization of which variables "go together". Variables that possibly represent the same underlying construct point in the same direction.  

biplot(pca)

#Calculate values for each student that represent these your composite variables and then create a new correlogram showing their relationship to mean_correct.

# Multiply pca$rotation to generate composite variables values
pca$rotation
##                               PC1         PC2         PC3        PC4
## prior_prob_count      -0.26034140  0.45818753 -0.40090679 -0.6897642
## prior_percent_correct  0.16840319  0.81617867  0.09267306  0.2640040
## problems_attempted    -0.45568733  0.31685183  0.36387724  0.3168141
## mean_hint             -0.63337594 -0.12501620 -0.08008842 -0.1122586
## mean_attempt          -0.54200011 -0.08510858 -0.04585364  0.3108682
## mean_confidence        0.03581325  0.02547483 -0.83051917  0.4948890
##                                PC5         PC6
## prior_prob_count      -0.007142834 -0.29280482
## prior_percent_correct  0.298843852  0.37134715
## problems_attempted    -0.592336569 -0.32911025
## mean_hint             -0.102302115  0.74412634
## mean_attempt           0.697232132 -0.33781385
## mean_confidence       -0.251357022 -0.01452143
DR1 <- t(pca$rotation)
DR2 <- pca$x %*% DR1
DR2 <- cbind(DR2, D1$mean_correct)
colnames(DR2)[7] <- "mean_correct"
DR2
##        prior_prob_count prior_percent_correct problems_attempted
##   [1,]       1.20419533          0.2031642901        -0.58392740
##   [2,]       2.79864517          0.7728564289         1.24541462
##   [3,]       3.04924632         -0.2819853646         0.12748339
##   [4,]       3.07117393          0.2505638372         0.63563395
##   [5,]       3.23719719         -0.9341716428        -0.38066717
##   [6,]       3.60056886         -0.0777519126         0.12748339
##   [7,]       1.52684432         -1.4391239118         0.12748339
##   [8,]       1.58636209         -0.8732812763         0.02585327
##   [9,]       0.82516109         -0.1507673786         0.94052428
##  [10,]       3.00852364         -0.3745407416         0.22911350
##  [11,]       2.94587335         -0.3508804396         0.22911350
##  [12,]       2.73599488          0.0060658363         1.24541462
##  [13,]       3.32177508          0.6305083792         0.02585327
##  [14,]       2.91141569          0.1105562402         0.22911350
##  [15,]       2.90201815         -0.5967851476         0.73726406
##  [16,]       1.06636470         -1.6764985926         1.04215439
##  [17,]       1.03503955         -1.4465656395         0.33074361
##  [18,]       0.72805314         -0.3288082708        -0.78718762
##  [19,]       3.76972464          0.5406904827        -0.58392740
##  [20,]       3.50972594          0.7119180359        -0.58392740
##  [21,]       4.08610860          0.5854553918        -0.58392740
##  [22,]       1.81816816          0.4203048814        -0.38066717
##  [23,]       1.36082105         -0.1549696708        -0.78718762
##  [24,]       3.60996641          0.4357239464        -0.38066717
##  [25,]       2.46973116          0.2562787399        -0.58392740
##  [26,]       3.14948679          0.3906768531        -0.68555751
##  [27,]       2.55430905         -0.3512238918        -0.38066717
##  [28,]       3.46587074          0.6254812029        -0.58392740
##  [29,]       3.72273692          0.0511930177        -0.27903706
##  [30,]       0.41480170          1.1661488779        -0.58392740
##  [31,]       0.44299433          0.1798578779         0.22911350
##  [32,]       0.41480170         -0.3092547292        -0.68555751
##  [33,]       0.41793421          0.3961841769        -0.17740695
##  [34,]       0.38034404          0.4529418210        -0.17740695
##  [35,]       0.40540415          0.7306246730        -0.17740695
##  [36,]       0.42106673          0.9329428935        -0.38066717
##  [37,]       0.41480170          1.5349997778        -0.58392740
##  [38,]       0.41480170          1.0186085135         0.22911350
##  [39,]       0.39287410         -1.4248636263        -0.27903706
##  [40,]       0.39600661          0.3820584287        -0.68555751
##  [41,]       0.41793421          0.3777878513        -0.78718762
##  [42,]       0.41793421          1.4815672433        -0.17740695
##  [43,]       0.41480170          1.0554936065         2.26171574
##  [44,]       0.55576485         -1.0117366847         1.85519529
##  [45,]       0.14853797         -1.0473100666         1.24541462
##  [46,]       0.39287410          0.7903171665         1.65193507
##  [47,]       0.41480170          1.2214765100        -0.58392740
##  [48,]       0.44299433          0.6848301708        -0.38066717
##  [49,]       0.40227164          0.1120216757         0.53400383
##  [50,]       0.34588638         -0.0286926072        -0.27903706
##  [51,]       0.38974158         -0.1356110757        -0.78718762
##  [52,]       0.45552438         -0.5859377817         0.33074361
##  [53,]       0.41793421          1.1136407744         2.46497596
##  [54,]       0.32395878          0.1869446955         0.73726406
##  [55,]       0.41480170          0.4284470707         0.02585327
##  [56,]       0.41793421          0.5985437297         0.63563395
##  [57,]       0.28010358          0.5196727313        -0.78718762
##  [58,]       0.41480170          0.3915619851         0.73726406
##  [59,]       0.40540415          0.7677899022         0.12748339
##  [60,]       0.41480170          0.8710681565        -0.38066717
##  [61,]       0.31456124         -1.0130948149         1.95682540
##  [62,]       0.41793421          0.8009032824         0.12748339
##  [63,]       0.10781528         -0.7491709099        -0.68555751
##  [64,]       0.08588768          0.3318912184         0.33074361
##  [65,]       0.36781398          0.8942778330        -0.58392740
##  [66,]       0.39913913         -1.1447536419        -0.78718762
##  [67,]       0.13914043         -0.0372569349        -0.78718762
##  [68,]       0.46805444          0.3963614852         1.34704473
##  [69,]       0.41793421          0.6905253432        -0.78718762
##  [70,]       0.43359678          0.3826146053         0.43237372
##  [71,]       0.41166918          0.6827405010         0.43237372
##  [72,]       0.44612684          0.2388685538         4.39594809
##  [73,]       0.42419924          0.3687502539         3.27801686
##  [74,]       0.45552438          0.5749104981        -0.17740695
##  [75,]       0.44925936         -1.1201856150        -0.38066717
##  [76,]       0.41793421          0.5249584344        -0.78718762
##  [77,]       0.41480170          1.1477063314         2.46497596
##  [78,]       0.41480170          0.4653321636         1.75356518
##  [79,]       0.41480170          1.5534423243         0.43237372
##  [80,]       0.42419924          0.1307908361         2.05845551
##  [81,]       0.42733176          0.2090633641         0.22911350
##  [82,]       0.41480170          0.5575448886        -0.17740695
##  [83,]       0.43046427          0.7605044092        -0.58392740
##  [84,]       0.41480170          0.0042685314        -0.78718762
##  [85,]       0.43046427          0.5601533883        -0.27903706
##  [86,]       0.60275256         -1.7227268233        -0.68555751
##  [87,]       0.42419924          0.7714508155        -0.58392740
##  [88,]       0.48684953          0.4400119933         2.26171574
##  [89,]       0.42419924         -0.6380011465         0.53400383
##  [90,]       0.32709129         -0.2637963536        -0.38066717
##  [91,]       0.38660907          0.8206237607        -0.68555751
##  [92,]       0.43046427          0.8697867895         1.24541462
##  [93,]       0.38974158          0.3913728299         3.17638675
##  [94,]       2.08129937         -0.1556461815        -0.58392740
##  [95,]       2.24419012          0.2924004370        -0.17740695
##  [96,]      -0.60013297         -3.1204967435        -0.27903706
##  [97,]       0.21118826         -0.5635735490        -0.27903706
##  [98,]       0.49624707          0.7297043085         0.12748339
##  [99,]       0.62781268          0.2826480648        -0.58392740
## [100,]       0.92540155         -0.2108944887         2.87149641
## [101,]       0.56202988         -0.8486888141        -0.58392740
## [102,]       0.69359548          0.3065823666         0.02585327
## [103,]       0.67480039          0.6335005697        -0.38066717
## [104,]       0.77817337         -0.3203776657        -0.27903706
## [105,]       0.55889736         -0.4588719548        -0.58392740
## [106,]       0.49937959         -0.4239022696        -0.78718762
## [107,]       0.65600531          1.3240947175        -0.27903706
## [108,]       0.59648753          0.1110411661        -0.58392740
## [109,]       0.60275256          0.5850879007         0.43237372
## [110,]       0.59335502         -0.1517432346         0.12748339
## [111,]       0.74058320         -0.0159374380         0.94052428
## [112,]       0.65913782          0.3853628098        -0.17740695
## [113,]       0.35841644          0.4587316738        -0.58392740
## [114,]       0.42106673          0.8778918964         0.63563395
## [115,]       0.38347655         -0.3937672216         1.85519529
## [116,]       0.34588638         -0.2824737990        -0.17740695
## [117,]       0.43046427          0.9426417048        -0.58392740
## [118,]       0.74058320          0.2618763547        -0.27903706
## [119,]      -0.46543485          0.0714183137        -0.78718762
## [120,]      -0.50929006          0.3473356876        -0.27903706
## [121,]      -0.48422994         -0.6737857554        -0.78718762
## [122,]      -0.48736246         -0.0663161175        -0.78718762
## [123,]      -0.60326549         -0.6549892897        -0.78718762
## [124,]       3.06177638          0.6842316710        -0.58392740
## [125,]       0.40227164          0.0002430043         1.24541462
## [126,]      -0.08640061         -0.9099547752        -0.27903706
## [127,]      -0.32760422         -1.7305907555        -0.58392740
## [128,]      -0.39651954         -0.9325862181        -0.78718762
## [129,]      -0.55627777         -2.0181241980        -0.78718762
## [130,]      -0.55941029          0.5484296106        -0.78718762
## [131,]      -0.54061520         -1.6315653815        -0.68555751
## [132,]      -0.39025451         -0.6273559352        -0.58392740
## [133,]      -0.61266303          0.4532127747        -0.27903706
## [134,]      -0.41844714         -1.6566782518        -0.78718762
## [135,]      -0.70350595         -2.0608467005        -0.88881773
## [136,]      -0.51242257         -0.9305153684        -0.38066717
## [137,]      -0.58760292         -1.0578329105        -0.68555751
## [138,]      -0.59386795         -0.3718623198        -0.78718762
## [139,]      -0.64398818         -0.3332300403        -0.68555751
## [140,]      -0.44663977         -0.2705703715        -0.27903706
## [141,]      -0.48109743         -0.7133440660        -0.58392740
## [142,]      -0.46856737         -0.6116488728        -0.68555751
## [143,]      -0.37459194          0.0614277908        -0.38066717
## [144,]      -0.23989382         -0.5215879518        -0.38066717
## [145,]      -0.40591708         -0.8945343230         0.02585327
## [146,]      -0.48109743         -0.7788809715         0.12748339
## [147,]      -0.50929006          0.8105479602         0.12748339
## [148,]      -0.34013428         -1.1303591273        -0.58392740
## [149,]      -0.34639931          0.0456155296        -0.58392740
## [150,]      -0.45290480         -0.0693492293        -0.58392740
## [151,]      -0.49049497         -0.2557882686         1.14378451
## [152,]      -0.56254280         -0.2393446165        -0.58392740
## [153,]      -0.53121766         -0.1344347754        -0.27903706
## [154,]      -0.15218341         -0.0304918353        -0.58392740
## [155,]       0.31456124          0.6113608423        -0.58392740
## [156,]       0.43359678          1.2910469066        -0.38066717
## [157,]      -0.30567662         -0.3201226622        -0.58392740
## [158,]      -0.42157965         -0.1035006879         0.12748339
## [159,]      -0.29314656         -1.2635492162        -0.78718762
## [160,]      -0.14591838         -0.1103447233        -0.38066717
## [161,]      -0.10519570          0.4007832547         0.02585327
## [162,]       0.41793421          0.5433547601         0.12748339
## [163,]      -0.25242387          0.6487607208        -0.58392740
## [164,]       0.01070734          0.6531855951        -0.38066717
## [165,]      -0.35266434         -0.9904837775         0.22911350
## [166,]      -0.12712330          0.3436933301        -0.58392740
## [167,]      -0.37459194         -0.6926954572         0.43237372
## [168,]      -0.25242387          0.2123203792        -0.58392740
## [169,]      -0.02688284         -1.8198211698        -0.78718762
## [170,]      -0.33073673         -0.3332300403        -0.78718762
## [171,]      -0.49989251         -1.1226405665         0.02585327
## [172,]      -0.44350725         -1.7302230882        -0.88881773
## [173,]      -0.40278457         -1.8360820819        -0.78718762
## [174,]      -0.49049497         -0.8955246317        -0.78718762
## [175,]      -0.45603731         -1.5871694168        -0.58392740
## [176,]      -0.55001274         -1.6789210787        -0.68555751
## [177,]      -0.46856737         -0.9913109224        -0.68555751
## [178,]      -0.48109743          0.0567145226        -0.78718762
## [179,]      -0.58447040         -1.1090542226        -0.68555751
## [180,]      -0.09579815          0.8927283383        -0.88881773
## [181,]      -0.03314786          0.9692837450        -0.58392740
## [182,]      -0.18037604          0.9654088668         0.73726406
## [183,]      -0.42784468         -0.0970939776        -0.68555751
## [184,]      -0.44037474          0.5769464455        -0.58392740
## [185,]      -0.42471217         -0.1920736391        -0.78718762
## [186,]      -0.61579555          0.4539726252        -0.68555751
## [187,]      -0.36832691         -0.5663288588        -0.38066717
## [188,]      -0.50929006         -0.7394315738        -0.68555751
## [189,]      -0.52495263          0.2135349657        -0.58392740
## [190,]      -0.38398948          0.1313028471        -0.27903706
## [191,]      -0.44977228         -0.4114445739        -0.17740695
## [192,]      -0.42157965         -1.4762736459        -0.68555751
## [193,]      -0.49049497         -0.6429971199        -0.58392740
## [194,]      -0.55314526         -1.1662114195        -0.88881773
## [195,]      -0.47483240          0.1754282985        -0.68555751
## [196,]      -0.40278457         -1.5681940134        -0.78718762
## [197,]      -0.49362748         -0.1429302966        -0.38066717
## [198,]      -0.50615754          0.2419823042         0.02585327
## [199,]      -0.43724222          1.6532662607        -0.58392740
## [200,]      -0.31507416         -0.2998657963        -0.38066717
## [201,]       0.11094780         -0.4466241209        -0.58392740
## [202,]      -0.02061781         -0.0030657588        -0.38066717
## [203,]       0.34901889         -0.0616256478         0.12748339
## [204,]      -0.64398818          0.1561121588        -0.78718762
## [205,]       0.51190964          0.9525764646        -0.58392740
## [206,]       0.49624707         -0.1185657749        -0.88881773
## [207,]      -0.03314786          0.9980685828        -0.88881773
## [208,]      -0.05194295          0.0838899629        -0.68555751
## [209,]      -0.51868760          0.2539805957         2.76986630
## [210,]      -0.77555378         -0.6186796559         1.75356518
## [211,]      -0.56254280          0.8702103592         0.22911350
## [212,]      -0.58133789          0.4007832547         1.44867484
## [213,]      -0.77242127         -0.8547658056        -0.27903706
## [214,]      -0.55001274         -0.1293374567         0.83889417
## [215,]      -0.65965075         -0.9671506106         0.22911350
## [216,]      -0.62206058          1.1402742633         3.98942764
## [217,]      -0.54061520          0.1048101492         0.53400383
## [218,]      -0.55941029          0.4640602654         2.46497596
## [219,]      -0.55941029          0.4640602654        -0.17740695
## [220,]      -0.33386925         -1.0418529420         1.14378451
## [221,]      -0.71603601         -1.7318229436         2.46497596
## [222,]      -0.65651823         -0.2545857566         0.33074361
## [223,]      -0.53435017         -0.1593847828         1.34704473
## [224,]      -0.61579555          0.4273779363        -0.58392740
## [225,]      -0.51242257          0.2208780347        -0.17740695
## [226,]      -0.56880783          0.6629308621        -0.27903706
## [227,]      -0.69097589          0.7677899022         0.83889417
## [228,]      -0.51555509          0.8549994017         0.12748339
## [229,]      -0.55941029          1.1390150195         0.94052428
## [230,]      -0.53435017          0.3814671149        -0.38066717
## [231,]      -0.53121766          0.0184846660         0.73726406
## [232,]      -0.52808514         -0.1099991939         1.75356518
## [233,]      -0.52808514          0.3440296476         2.56660607
## [234,]      -0.69097589          0.1153336391         0.43237372
## [235,]      -0.60013297         -0.0455761829         0.02585327
## [236,]      -0.41531462         -0.3028761096         0.63563395
## [237,]      -0.55941029         -0.3796331791         2.46497596
## [238,]      -0.50302503          0.0687296230         0.12748339
## [239,]      -0.37459194          0.1242713961         1.24541462
## [240,]      -0.53121766         -0.6696527981         0.22911350
## [241,]      -0.47169988         -0.7002366878         0.43237372
## [242,]      -0.54374771          0.0816470385         0.43237372
## [243,]      -0.55314526         -0.8156994527         0.02585327
## [244,]      -0.24302633         -0.9657308601         5.10735887
## [245,]      -0.51868760          0.0337766072        -0.58392740
## [246,]      -0.50929006         -1.4698816966         1.95682540
## [247,]      -0.75049367         -1.9988755949        -0.78718762
## [248,]      -0.44977228          0.7316908861         0.02585327
## [249,]      -0.42471217          1.1065652679        -0.17740695
## [250,]      -0.82254150         -2.6576054769        -0.58392740
## [251,]      -0.74109612         -0.2952638339         0.22911350
## [252,]      -0.55001274         -0.5371226239        -0.58392740
## [253,]      -0.68157835         -0.0579750547        -0.27903706
## [254,]      -0.56567532          0.4223718775        -0.68555751
## [255,]      -0.60953052         -0.3487373630        -0.68555751
## [256,]      -0.57194034          1.3514631225        -0.48229729
## [257,]      -0.49989251          0.9201322843        -0.58392740
## [258,]      -0.58447040          0.0059026872        -0.27903706
## [259,]      -0.60013297         -1.0374860400        -0.68555751
## [260,]      -0.74109612          0.8437223150        -0.68555751
## [261,]      -0.74109612         -1.8771890431        -0.38066717
## [262,]      -0.68157835         -1.3960201261        -0.78718762
## [263,]      -0.74109612          0.2742292406        -0.27903706
## [264,]      -0.74109612         -0.4850948587        -0.38066717
## [265,]      -0.74109612          1.8561544425         0.43237372
## [266,]      -0.74109612         -0.9280339191        -0.68555751
## [267,]      -0.74109612         -2.3201280962        -0.78718762
## [268,]      -0.74109612          0.0211212050        -0.88881773
## [269,]      -0.74109612          1.2233843647        -0.78718762
## [270,]      -0.70977098         -0.8696243707        -0.38066717
## [271,]      -0.51868760          0.1805792662         2.66823619
## [272,]       0.38660907          0.4055005673         4.29431798
## [273,]      -0.71916852          0.4007832547        -0.38066717
## [274,]      -0.63459063         -0.0943844410         3.27801686
## [275,]      -0.77555378         -0.6186796559        -0.58392740
## [276,]      -0.78808384         -0.3856595603         0.63563395
## [277,]      -0.67531332         -0.5534340288         2.16008563
## [278,]      -0.79434887          1.0124609980         0.33074361
## [279,]      -0.81001144          1.1872260697        -0.68555751
## [280,]      -0.75049367          0.2596268535         0.94052428
## [281,]      -0.70350595         -2.2398743342         0.33074361
## [282,]      -0.78181881         -0.5167333641        -0.78718762
## [283,]      -0.76928875          0.0337766072        -0.48229729
## [284,]      -0.82254150         -2.6576054769        -0.58392740
## [285,]      -0.78808384          0.1386356472        -0.68555751
## [286,]      -0.80061390          0.0337766072        -0.88881773
## [287,]      -0.80374641         -1.0264648157        -0.07577684
## [288,]      -0.70350595         -0.6286256314        -0.27903706
## [289,]      -0.66591578          0.5738996003        -0.78718762
## [290,]      -0.59073543          1.2825524717        -0.58392740
## [291,]      -0.82254150          2.2358164921        -0.58392740
## [292,]      -0.81314396          2.2358164921         0.12748339
## [293,]      -0.74736115          1.1483893895        -0.58392740
## [294,]      -0.79434887          1.6241387487         2.16008563
## [295,]      -0.65338572         -0.0821202312        -0.68555751
## [296,]      -0.60639800         -0.2108944887        -0.58392740
## [297,]      -0.62519309          0.3451761837        -0.17740695
## [298,]      -0.66591578          0.5738996003         0.12748339
## [299,]      -0.69724092          0.5288088260        -0.58392740
## [300,]      -0.61892806         -0.2468755342         0.22911350
## [301,]      -0.70350595          0.0874848958        -0.58392740
## [302,]      -0.68784338         -0.1577051255         0.63563395
## [303,]      -0.80061390         -1.4342499828        -0.58392740
## [304,]      -0.68471086          0.8302591166        -0.58392740
## [305,]      -0.72856607          0.4563903256         0.63563395
## [306,]      -0.71916852         -0.8225722394        -0.38066717
## [307,]      -0.54374771         -0.0779210659        -0.58392740
## [308,]      -0.71290349         -0.6616044084        -0.78718762
## [309,]      -0.78808384         -1.9585451903        -0.78718762
## [310,]      -0.80374641         -1.8420351499        -0.78718762
## [311,]      -0.81314396          1.0124609980         0.02585327
## [312,]      -0.82880653          2.2358164921        -0.78718762
## [313,]      -0.80374641         -0.2108944887        -0.78718762
## [314,]      -0.81314396          1.0124609980        -0.38066717
## [315,]      -0.81940898          2.2358164921        -0.58392740
## [316,]      -0.82880653          2.2358164921        -0.68555751
## [317,]      -0.81940898          2.2358164921        -0.68555751
## [318,]      -0.81314396          1.0124609980        -0.68555751
## [319,]      -0.82880653          2.2358164921        -0.07577684
## [320,]      -0.79748138          0.2339620493        -0.68555751
## [321,]      -0.81940898          2.2358164921         0.94052428
## [322,]      -0.81314396         -0.2108944887        -0.78718762
## [323,]      -0.81627647         -2.1682632778        -0.78718762
## [324,]      -0.82880653          2.2358164921        -0.68555751
## [325,]      -0.82880653          2.2358164921        -0.88881773
## [326,]      -0.81314396          1.0124609980         0.02585327
## [327,]      -0.82880653          2.2358164921         1.14378451
## [328,]      -0.76928875          1.1347965496        -0.78718762
## [329,]      -0.82880653          2.2358164921         1.55030495
## [330,]      -0.78495133          0.7677899022         0.02585327
## [331,]      -0.82567401         -1.4342499828         0.33074361
## [332,]      -0.82254150         -2.6576054769        -0.68555751
## [333,]      -0.82254150         -0.2108944887        -0.38066717
## [334,]      -0.81314396         -2.6576054769        -0.88881773
## [335,]      -0.80061390          0.7677899022        -0.78718762
## [336,]      -0.82880653          2.2358164921        -0.78718762
## [337,]      -0.81314396         -1.4342499828        -0.38066717
## [338,]      -0.80374641         -3.4731758038         1.34704473
## [339,]      -0.81314396          1.0124609980         0.63563395
## [340,]      -0.79434887          0.4007832547        -0.68555751
## [341,]      -0.56880783         -0.1235119529        -0.68555751
## [342,]      -0.76928875          2.2358164921        -0.58392740
##            mean_hint  mean_attempt mean_confidence mean_correct
##   [1,] -0.6979846340 -3.882824e-01     0.442731791    1.0000000
##   [2,]  2.0634619537 -1.074970e-01    -0.623140674    0.4545455
##   [3,]  0.9926969503  5.102308e-01     0.072767766    0.6363636
##   [4,] -0.0005784805 -1.566345e-01    -0.111582708    0.7500000
##   [5,]  1.9883205499  8.471732e-01    -0.884564066    0.3333333
##   [6,]  1.7816816896 -5.133996e-02     0.928638778    0.5454545
##   [7,]  1.8943937953  8.471732e-01    -0.574895672    0.3636364
##   [8,]  3.5174481162  1.959083e+00     0.005925587    0.3000000
##   [9,]  2.1732079513  5.220533e-01     0.133943133    0.4210526
##  [10,]  2.9181954213  4.353547e-01     0.798065421    0.2500000
##  [11,]  1.5750428293  1.264908e-01    -0.908988769    0.5833333
##  [12,]  1.9507498481  2.856025e-01     0.169099443    0.5000000
##  [13,]  0.5418485278 -1.411913e-01     1.289136198    0.8000000
##  [14,]  1.9883205499  3.324001e-01     0.313662707    0.4166667
##  [15,]  2.1463385019  2.657823e-01    -0.825356651    0.3529412
##  [16,]  3.0835065096  9.089460e-01    -0.208964629    0.2500000
##  [17,]  1.8770534713  1.215634e+01     1.871325461    0.3076923
##  [18,] -0.6979846340 -3.882824e-01     0.764982791    0.5000000
##  [19,] -0.6979846340 -3.882824e-01    -2.054005135    1.0000000
##  [20,] -0.6979846340 -3.882824e-01     0.104806442    1.0000000
##  [21,] -0.6979846340 -3.882824e-01     1.318215830    1.0000000
##  [22,]  0.3352096675 -1.823731e-01     0.717640176    0.8333333
##  [23,] -0.6979846340 -1.006010e+00    -0.885664528    0.5000000
##  [24,]  0.3352096675 -1.823731e-01    -1.399394468    0.8333333
##  [25,] -0.6979846340 -3.882824e-01     1.077334554    1.0000000
##  [26,]  1.3684039690  2.353614e-02     0.847683489    0.3333333
##  [27,]  0.3352096675 -1.823731e-01     0.432499686    0.8333333
##  [28,] -0.6979846340 -3.882824e-01    -0.189852854    1.0000000
##  [29,] -0.6979846340 -2.117887e-01     0.178845134    0.8571429
##  [30,] -0.6979846340 -3.882824e-01     0.015034046    1.0000000
##  [31,] -0.6979846340 -1.823731e-01     0.032678602    0.9166667
##  [32,]  1.3684039690  2.353614e-02    -1.014977930    0.3333333
##  [33,] -0.5430054888  7.501346e-02    -0.458811470    0.6250000
##  [34,] -0.6979846340 -3.882824e-01     1.500990867    1.0000000
##  [35,] -0.6979846340 -3.882824e-01    -1.044516163    1.0000000
##  [36,] -0.6979846340 -1.823731e-01     0.597553478    0.8333333
##  [37,] -0.6979846340 -3.882824e-01    -0.361141379    1.0000000
##  [38,] -0.0780680531  5.383093e-01    -0.232553661    0.8333333
##  [39,]  0.8960865740  3.176922e-01     1.790719614    0.7142857
##  [40,] -0.6979846340 -8.001010e-01    -1.054436396    0.6666667
##  [41,] -0.6979846340  2.294454e-01    -0.593676383    0.5000000
##  [42,] -0.6979846340 -3.882824e-01    -0.147816370    1.0000000
##  [43,]  1.3167442540 -4.081050e-02     1.012952253    0.5312500
##  [44,]  3.1986338746  1.411986e-01     1.564628993    0.1785714
##  [45,]  3.6977874852 -5.133996e-02    -0.826882861    0.1363636
##  [46,]  0.0173037286  3.244805e-01     0.226448849    0.6153846
##  [47,] -0.6979846340 -3.882824e-01     0.422765451    1.0000000
##  [48,] -0.6979846340 -1.823731e-01     0.652934539    0.8333333
##  [49,]  1.1204373367  2.706273e-01     0.404359382    0.6666667
##  [50,]  0.0104914585 -2.117887e-01     1.543610437    0.8571429
##  [51,] -0.6979846340 -3.882824e-01     1.993336172    0.5000000
##  [52,]  3.0215148515  7.521382e-01     0.287412342    0.2307692
##  [53,] -0.5156562279 -2.429347e-01    -0.782443596    0.9117647
##  [54,]  0.0313289906  4.111301e-01     0.484401950    0.8235294
##  [55,] -0.0780680531  7.236277e-01    -1.885373029    0.7000000
##  [56,]  0.0769110921  3.066614e-01     1.805165171    0.8125000
##  [57,] -0.6979846340  2.294454e-01    -1.655857534    0.5000000
##  [58,] -0.6979846340  1.931085e-01    -0.371933646    0.8235294
##  [59,] -0.2471362115 -2.759683e-01    -0.324971261    0.9090909
##  [60,] -0.4913457737 -1.823731e-01     1.038057704    0.8333333
##  [61,]  0.8838714690  2.933483e-01    -0.215883165    0.6896552
##  [62,] -0.6979846340 -2.759683e-01    -1.789510157    0.9090909
##  [63,]  1.3684039690 -3.882824e-01     1.044589539    0.3333333
##  [64,] -0.1257539439  5.620681e-01     0.484651062    0.5384615
##  [65,] -0.6979846340 -3.882824e-01    -0.122409339    1.0000000
##  [66,] -0.6979846340 -1.006010e+00     0.086864711    0.5000000
##  [67,]  2.4015982705  2.294454e-01    -1.054943295    0.5000000
##  [68,]  2.2129280068  2.619784e+00    -1.174236340    0.3043478
##  [69,] -0.6979846340 -3.882824e-01     0.219350280    0.5000000
##  [70,] -0.2551870762  4.059391e-01     1.050813348    0.7857143
##  [71,]  0.1876104816  1.411986e-01     0.306574045    0.8571429
##  [72,]  0.3780969781  2.177902e-01     0.148046371    0.7169811
##  [73,] -0.1666275647  4.353547e-01    -2.449975405    0.7380952
##  [74,] -0.6979846340  6.927413e-01     0.364166677    0.7500000
##  [75,] -0.6979846340  2.294454e-01    -0.271430925    0.8333333
##  [76,] -0.6979846340 -1.006010e+00     1.939100637    0.5000000
##  [77,]  2.0734071395  1.567716e-01    -1.014679738    0.4117647
##  [78,]  0.0826510605  1.150514e-01    -1.154347599    0.8518519
##  [79,] -0.6979846340 -3.529508e-02    -1.053214141    0.9285714
##  [80,]  0.2525541234  3.118091e-01     1.081562866    0.7333333
##  [81,] -0.6979846340 -3.882824e-01     1.467600917    1.0000000
##  [82,] -0.5430054888 -2.338504e-01    -1.682909308    0.8750000
##  [83,] -0.6979846340 -3.882824e-01     1.180363618    1.0000000
##  [84,] -0.6979846340  8.471732e-01    -1.535046724    0.5000000
##  [85,]  0.1876104816  1.376654e+00    -0.023192945    0.7142857
##  [86,]  1.3684039690 -3.882824e-01     1.224481751    0.3333333
##  [87,] -0.6979846340 -3.882824e-01    -0.894903269    1.0000000
##  [88,] -0.4267711299 -4.081050e-02    -1.490324763    0.8437500
##  [89,]  2.9388593073  4.353547e-01     0.330968533    0.2666667
##  [90,]  1.1617651087  2.294454e-01     1.192050495    0.5000000
##  [91,] -0.6979846340 -3.882824e-01     1.568817347    0.6666667
##  [92,] -0.6979846340 -1.636541e-01    -0.741710140    0.8636364
##  [93,]  1.1466451921  6.371356e-02     0.168515227    0.6341463
##  [94,] -0.6979846340 -3.882824e-01    -0.580744149    1.0000000
##  [95,] -0.6979846340 -3.882824e-01     0.423579168    0.7500000
##  [96,]  1.9588007128  2.082629e+00    -0.298812840    0.2857143
##  [97,]  0.1876104816  3.176922e-01     1.199358896    0.7142857
##  [98,] -0.6979846340  1.732883e-01     0.244864026    0.8181818
##  [99,] -0.6979846340 -3.882824e-01     0.812544846    1.0000000
## [100,]  0.7702388471  8.796852e-01    -0.412183797    0.6842105
## [101,] -0.6979846340 -3.882824e-01    -0.656670064    1.0000000
## [102,] -0.6979846340  1.058999e-01     2.228673144    0.8000000
## [103,] -0.6979846340 -1.823731e-01     0.427377322    0.8333333
## [104,]  0.8960865740 -2.117887e-01     0.766280796    0.7142857
## [105,] -0.6979846340 -3.882824e-01     0.251137279    1.0000000
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## [108,] -0.6979846340 -3.882824e-01    -0.786678377    1.0000000
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## [117,] -0.6979846340 -3.882824e-01    -0.438661284    1.0000000
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## [134,] -0.6979846340 -1.006010e+00     1.179783244    0.5000000
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## [137,] -0.6979846340  2.353614e-02    -0.911002946    0.3333333
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## [141,] -0.6979846340 -3.882824e-01     0.170897785    1.0000000
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## [152,] -0.6979846340 -3.882824e-01    -0.023387958    1.0000000
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## [154,] -0.6979846340 -3.882824e-01    -0.255437787    1.0000000
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## [168,] -0.6979846340 -3.882824e-01     0.327260577    1.0000000
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## [184,] -0.6979846340 -3.882824e-01     0.829405016    1.0000000
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## [204,] -0.6979846340 -1.006010e+00    -1.185593846    0.5000000
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## [224,] -0.6979846340 -3.882824e-01    -0.836894635    1.0000000
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## [329,]  1.8808683426  2.047363e-01     1.125604091    0.4800000
## [330,] -0.0780680531 -1.411913e-01    -0.520862974    0.9000000
## [331,]  0.7325920912  2.769629e-01     1.076376295    0.6153846
## [332,]  1.7816816896  1.258992e+00    -1.989829184    0.3333333
## [333,]  0.3352096675 -3.882824e-01     0.722060612    0.8333333
## [334,] -0.6979846340 -1.623738e+00    -0.583824390    0.0000000
## [335,] -0.6979846340 -1.006010e+00     0.737555030    0.5000000
## [336,] -0.6979846340 -1.006010e+00    -1.154664327    0.5000000
## [337,] -0.0780680531  8.471732e-01     1.796145003    0.5000000
## [338,] -0.6979846340  4.144130e-02     0.934233541    0.7826087
## [339,] -0.0005784805  9.243892e-01     0.642481802    0.5625000
## [340,]  0.1285708072  2.353614e-02     0.567818716    0.6666667
## [341,]  1.3684039690  2.353614e-02     0.606821017    0.3333333
## [342,] -0.6979846340 -3.882824e-01    -0.606413189    1.0000000
# install.packages("corrgram")
library("corrgram")
## Registered S3 method overwritten by 'seriation':
##   method         from 
##   reorder.hclust gclus

# Plot
corrgram(DR2)

# Part III